Mixed integral equation
An integral equation that, in the one-dimensional case, has the form
(1) |
where is the unknown and is a given continuous function on , , , are given points, and , are given continuous functions on the rectangle . If
where the are positive constants, then (1) can be written as
(2) |
where the new integration symbol, with an arbitrary finite integrable function, is defined by (see [1]):
The theory of Fredholm equations (cf. Fredholm equation) and, in the case of a symmetric kernel, the theory of integral equations with symmetric kernel (cf. Integral equation with symmetric kernel), is valid for equation (2).
In the case of multi-dimensional mixed integral equations, the unknown function can be part of the integrands of integrals over manifolds of different dimensions. For example, in the two-dimensional case the integral equation may have the form
where is some domain in the plane, is its boundary, and are fixed points in . This equation may also be written as
if the function and the volume element are correspondingly defined. In this case, moreover, the theory of Fredholm integral equations remains valid.
References
[1] | A. Kneser, "Belastete Integralgleichungen" Rend. Circolo Mat. Palermo , 37 (1914) pp. 169–197 |
[2] | L. Lichtenstein, "Bemerkungen über belastete Integralgleichungen" Studia Math. , 3 (1931) pp. 212–225 |
[3] | N.M. Gunter, "Sur le problème des "Belastete Integralgleichungen" " Studia Math. , 4 (1933) pp. 8–14 |
[4] | V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) |
Mixed integral equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_integral_equation&oldid=18355